Are you trying to launch Firefox from the menu? Because that will continue launching Firefox as installed on your system, not the Nightly app you downloaded. As someone else mentioned, to integrate into menus there are a lot of other steps. But if you navigate to the folder you extracted and click or double click the file named 'firefox', it should run the downloaded version. I think but I'm not sure that because of the way Firefox works, it will run with your existing profile from the normal version of Firefox. Check in Help > About Firefox to make sure which version is running.

Since it’s Firefox, if you feel the need to use nightly builds, there’s an alternative way pf getting it. Add the Mozilla daily ppa (while adding ppa isn’t a promoted way to install apps on eOS by the devs) and install firefox-trunk from the apt command.